Hemispherical open resonator (OR) with the segment of the oversized circular waveguide is considered. The cavity is formed by cylindrical, conical and spherical surfaces, and only axial-symmetric modes are excited. The power and spectral characteristics of such cavity with a dielectric bead and a rod have been studied. The quasi-periodic behavior of those dependencies was found out. Their qualitative agreement with similar dependencies for a cylindrical cavity is shown. It was found that physical processes in the cavity and in the hemispherical ОR with the segment of the oversized circular waveguide are identical. Dielectric permittivity and loss-angle tangent measurements have been carried out in millimeter wavelength range for the Teflon and Plexiglas samples having the shape of the bead as well as for the fused quartz and glass samples of the rod shape. It was found out that such a resonant system allows measuring samples with high losses, that is, especially important for quality control of food stuffs and analysis of biological liquids. Energy analysis of the ОR with the segment of the oversized rectangular waveguide has been performed. Basic possibility to apply such a resonant system for measurement of the dielectric permittivity of cylindrical samples with high losses has been shown as well.